ﻻ يوجد ملخص باللغة العربية
We study conditional independence (CI) models in statistical theory, in the case of discrete random variables, from the point of view of algebraic geometry and matroid theory. Any CI model with hidden random variables corresponds to a variety defined by certain determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We show that any CI variety, and more generally any hypergraph variety, admits a matroid stratification. Our main motivation for studying decompositions of CI varieties is the realizability problem: given a collection of CI relations, the goal is to determine the existence of random variables that satisfy these constraints and violates the rest. We show that the realization spaces of CI models and the matroid varieties in their decompositions are closely related. We use ideas from incidence geometry, in particular point and line configurations, to find minimal decompositions of general hypergraph varieties in terms of matroid varieties, which are not necessarily irreducible by Mnev--Sturmfels universality theorem, and may have arbitrary singularities. We focus on various families of hypergraph varieties for which we explicitly compute an irredundant irreducible decomposition. Our main findings in this direction are threefold: (1) we describe minimal matroids of such hypergraphs; (2) we prove that the varieties of these matroids are irreducible and their union is the hypergraph variety; and (3) we show that every such matroid is realizable over real numbers. Our decomposition strategy gives immediate matroid interpretations of the irreducible components of many families of CI varieties in algebraic statistics, and unravels the symmetric structures in CI varieties which hugely simplifies the computations.
Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be descr
This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces b
The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this varietys cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen yea
We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We introduce comb